Q. 57 - Precalculus Enhanced with Graphing Utilities

Question

Financial Planning A retired couple has up to \(50,000 to

invest. As their financial adviser, you recommend that they

place at least \)35,000 in Treasury bills yielding 1% and at

most $10,000 in corporate bonds yielding 3%.

(a) Using x to denote the amount of money invested in

Treasury bills and y the amount invested in corporate

bonds, write a system of linear inequalities that describes

the possible amounts of each investment.

(b) Graph the system and label the corner points.

Step-by-Step Solution

Answer

(a)The inequalities of the retired couple

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 50000\\ y\le 10000\\ x\ge 35000\end{array}\right.$

(b)

step 1. introduction

(a) the couple has total invested the total money in two different x and y of $50000$

we have first inequality $x+y\le 50000$

step 2.investments made

The couple made investment at least of 35000 in treasury bills so the inequality is $x\ge 35000$

another investment not more than 10000 in corporate bonds so the inequality is$y\le 10000$

both the investments are not zero so $x\ge 0$&$y\ge 0$

step 3.inequalities

(a):$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 50000\\ y\le 10000\\ x\ge 35000\end{array}\right.$

step 4.plotting the graph

since $x\ge 0$&$y\ge 0$ the graph lies on x-axis

we have already an inequality $x+y\le 50000$

by  using test point$(0,0)$ 

$$\begin{gathered} 0+0\le 50000 \\ 0\le 50000 \end{gathered}$$ 

step 5.tesing the lines

$$\begin{gathered} y\le 10000 \\ 0\le 10000 \end{gathered}$$substitute point $(0,0)$

the graph includes origin the shaded region

$$\begin{gathered} x\ge 35000 \\ 0\ge 35000 \end{gathered}$$substitute point $(0,0)$ 

the graph does not include  orgin

step 6.the resultant graph and corner points

The corner points are $$\begin{gathered} (35000,10000) \\ (40000,10000) \\ (35000,0) \\ (50000,0) \end{gathered}$$